Minus Infinity

Describe MinusInfinity here.

 | minusInfinity |
 minusInfinity := 0.
 [minusInfinity - 1 < minusInfinity] whileTrue: 
 [minusInfinity := minusInfinity - 1].
 ^minusInfinity

In the J language (wiki name JayLanguage) the expression is:

  minusInfinity =: __ (and they say that interpreters are slow!)

The word for minus infinity is the negative symbol (an underscore, to avoid confusion with negation) and the infinity symbol, which is also the underscore character. Thus:

]infinity =: _ NB. one underscore _

]minusInfinity =: __ NB. two underscores __ minusInfinity * 2 NB. twice minusInfinity is still minusInfinity __ minusInfinity * _1 NB. minusInfinity times minus one is Infinity (one underscore) _

minusInfinity + _ NB. but minusInfinity + infinity is not 0

_. NB. the symbol for Indeterminate

Mathematically in the context of the real numbers, -oo (minus infinity) is an unbounded number that is less than every real number.

As far as I know, this does not make sense. There is no number oo or a number -oo. It's just a symbol. IF I'm wrong, could someone explain how you add and subtract numbers if oo and -oo are considered numbers?

You're not wrong. Infinity is not a number. (More precisely: introducing infinity as a first-class element in any of the usual number systems would mess things up far worse than the value of doing so would help, so we don't do that.) But (1) it's possible to add infinity (or several infinities) as a second-class citizen, and sometimes this gets done for convenience; and (2) there are usable number systems with lots of infinities in them. (For instance, Conway's SurrealNumbers.)

Note that #2 is very different from introducing a "number" called "infinity" and another called "minus infinity". You need a vast number (oops) of different infinities.

Nothing is stopping you from only adding the two, it's just that it's not very useful to do so. It is common in complex analysis, though, to add a single infinity (plus, minus and imaginary) to the complex plane.

: Yes, that's true. But once you do that, you generally stop thinking about the elements of the complex plane (or, once you've added the point at infinity, the Riemann sphere) as numbers and switch into topologist mode. :-) When I said "You need ..." I meant "You need, if you want a system that's bearable to use ...".

By "second-class citizen" I mean that lots of the usual laws of arithmetic fail completely when you work with infinities, and some operations on them yield completely indeterminate results. It's best to think of what you're doing as a shorthand; trying to treat infinity as a "number" will almost always lead to much more confusion than it saves.

NonStandardAnalysis is simpler than the standard calculus and it is made so precisely by the introduction of the hyper-real number system. Let's say that H is a positive infinite number in the hyper-real system (R* from now on) then H^2 > 2*H > H+1 > H. There are as many infinite numbers as there are reals and as many infinitesimals between each real number as there are reals. R* is defined in such a way that every function over the reals is preserved so every last law of arithmetic is preserved by definition. The benefits of using R* come from using the std() function instead of epsilon-delta proofs when working with limits from scratch. The standard part of a finite hyper-real number is its real part. The standard part of an infinite number is undefined. The hyperreals are the intuitive way of creating calculus (and how it was invented) but it had to wait for mathematical logic to be rigorously defined. In the meantime, a horribly non-intuitive foundation for calculus has come to dominate.

A couple of quibbles.

You don't mean "every function over the reals is preserved" unless I don't understand what you mean by "function". What's true is that every first-order statement is true of R* iff it's true of R.
It's not really fair to say that the hyperreals are "how calculus was invented". IsaacNewton and GottfriedWilhelmLeibniz certainly thought in terms of infinitesimals, but it's no more true that they were "really" thinking of the hyperreals than that they were "really" thinking of the SurrealNumbers, or any other system that extends the reals by adding infinities and infinitesimals.

However, it seems that I (the author of the remarks about second-class citizens, to which exception seems to be being taken) was not clear enough above. In NSA, and in Conway's SurrealNumbers, and in various other systems, infinities are not second-class citizens. But these systems share the feature that they add not a single "infinity", nor a few infinities (say, one positive and one negative), but shedloads of them. There are good systems with lots of infinities in, but just adding in "infinity" and treating that as a number does not make one of them.

In the meantime, a horribly non-intuitive foundation for calculus has come to dominate.

Commented elsewhere, I don't think epsilon-delta is so bad. Thinking about it a bit more, it seems that for any of the results to make sense, you should use non-standard analysis, but in order to see why they should apply without the extra objects you should have the epsilon-delta framework handy. Whenever you can do something two different ways in math, both have at least the merit of illuminating the other. -- JoshuaGrosse

The way it was explained to me in calculus, there is no such thing as infinity; but one can work rationally with "numbers that grow arbitrarily large." Like, you can find the "limit" of "(x+1)/(x+2)" as x grows "arbitrarily large" is a value that gets "arbitrarily close" to being 1. So you say that the expression "goes to 1 as x approaches infinity."

In this way you can compare, contrast and compute values related to "infinity," because none of them actually ever become "infinite." -- JeffGrigg

That's the usual way. The thing is, it soon become mysterious why things that don't exist behave so much like normal numbers, why dy/dx acts exactly like a ratio, and so forth. All that mystery goes away when you add infinities and infinitesimals to the system, and all kind of results become much more obvious. But I would still say the standard formulism is handy as a way to know you can do that.

The phrase "grows arbitrarily large" perpetuates a notion of infinity as an ever shifting, ever growing object. In actuality, you're not dealing with infinity at all. You're never "approaching" infinity but the most common way for students to understand this is precisely that infinity is receding away from them. This is extremely confusing and dangerous when you get to set theory or any other theory which makes use of actual infinities. Better to do away with it and introduce infinities as actual numbers. And just as epsilon-delta might be handy when you don't want to resort to extensions of the real number system, so the extensions are handy when you want to show exactly what epsilon-delta does not mean. There is a reason why the Axiom of Infinity exists; no amount of mathematical induction will get you an infinite set from scratch.

Thinking of infinity as receding is misleading, but that's a problem with students and not inherent in the model. I don't know about the hyperreals per se, but in most extensions of the real numbers I've seen an unbounded sequence does approach infinity, in a perfectly valid topological sense: given any open interval around infinity, there will be some point after which the sequence is contained entirely within it. I'm not entirely sure why you bring up the AoI.

In NonStandardAnalysis, there isn't a single thing called "infinity". There are many different infinities. For most of them, it isn't true (e.g.) that the sequence 1,2,3,4,... approaches them in that sense. (Let w be an infinitely large hyperreal; then 2w is just as far from w as w is from 0.) But this may be a little misleading; "the sequence 1,2,3,4,..." isn't as well defined as you think it is. For instance, if you do nonstandard analysis "internally" then that sequence includes lots of infinite natural numbers. (Really.) And in that case it ultimately exceeds any hyperreal number you care to choose. On the other hand, if you insist on continuing to think about the ordinary natural numbers and real numbers at the same time as working with the hyperreals, then you need to be very careful that you don't mix the two in improper ways. (Any first-order statement that's true about the reals is also true about the hyperreals, but it won't usually be true if you interpret some of the things in it as referring to the ordinary reals and the others as referring to the hyperreals.) So, for instance, the sequence 1,2,3,4,... (if taken as ranging only over the ordinary natural numbers) has upper bounds, but no least upper bound, within the hyperreals. Ouch.

CategoryMath